To calculate the correlation coefficient $r$ between two variables (temperature $x$ and commute time $y$), we can use the Pearson correlation formula:

[ r = \frac{n\sum{xy} - (\sum{x})(\sum{y})}{\sqrt{[n\sum{x^2} - (\sum{x})^2][n\sum{y^2} - (\sum{y})^2]}} ]

Where:

• $n$ is the number of data points,
• $\sum{xy}$ is the sum of the products of corresponding $x$ and $y$ values,
• $\sum{x}$ and $\sum{y}$ are the sums of the $x$ and $y$ values, respectively,
• $\sum{x^2}$ and $\sum{y^2}$ are the sums of the squares of the $x$ and $y$ values.

Let's calculate the necessary components.

Given dataTo calculate the correlation coefficient $r$ between two variables (temperature $x$ and commute time $y$), we can use the Pearson correlation formula:

[ r = \frac{n\sum{xy} - (\sum{x})(\sum{y})}{\sqrt{[n\sum{x^2} - (\sum{x})^2][n\sum{y^2} - (\sum{y})^2]}} ]

Where:

• $n$ is the number of data points,
• $\sum{xy}$ is the sum of the products of corresponding $x$ and $y$ values,
• $\sum{x}$ and $\sum{y}$ are the sums of the $x$ and $y$ values, respectively,
• $\sum{x^2}$ and $\sum{y^2}$ are the sums of the squares of the $x$ and $y$ values.

Let's calculate the necessary components.

Given

• $x = [3, 0, 1, 2, 10]$
• $y = [59, 39, 50, 50, 31]$

We will now compute the correlation coefficient.The correlation coefficient $r$ for the given data is approximately $-0.555$, rounded to the nearest thousandth.

Would you like more details on the calculation, or do you have any questions?

Here are five related questions to expand your understanding:

1. What does a negative correlation coefficient indicate about the relationship between two variables?
2. How can the strength of the relationship between two variables be interpreted using the correlation coefficient?
3. What is the difference between correlation and causation in data analysis?
4. Can a non-linear relationship have a correlation coefficient close to 0?
5. How would you interpret an $r$-value of 1 or -1?

Tip: The correlation coefficient only measures linear relationships; non-linear relationships may require different methods for analysis.